Gradient-Based Optimization on G\"odel Logic as Discrete Local Search

A fundamental challenge in neurosymbolic systems is applying continuous gradient-based optimization to discrete logical domains. While fuzzy relaxations provide differentiability, they often lack a formal structural alignment with classical logic. In this work, we show that G\"odel semantics addresses this limitation through a homomorphism that maps its continuous interpretations to Boolean ones, allowing discrete variables to be encoded while maintaining full differentiability. Building on this foundation, we show that gradient-based optimization on G\"odel logic instantiates a discrete local search for Boolean satisfiability. Our formal analysis proves that each optimization step identifies and modifies a single variable within a unsatisfied clause, precisely mimicking the steps of a discrete solver. We identify local optima as the primary limitation of such dynamics and introduce the G\"odel Trick, a stochastic reparameterization technique designed to improve the exploration of the solution space. We further show a formal connection between this approach, probabilistic inference, and the Gumbel-Max trick. Experimental results on SAT benchmarks and the Visual Sudoku task validate our theoretical findings, demonstrating that our approach effectively navigates complex combinatorial landscapes and provides a solid foundation for differentiable discrete search.